6 Answers
6

Tidal locking (or captured rotation) occurs when the gravitational
gradient makes one side of an astronomical body always face another,
an effect known as synchronous rotation. For example, the same side of
the Earth's Moon always faces the Earth. A tidally locked body takes
just as long to rotate around its own axis as it does to revolve
around its partner. This causes one hemisphere constantly to face the
partner body. Usually, at any given time only the satellite is tidally
locked around the larger body, but if the difference in mass between
the two bodies and their physical separation is small, each may be
tidally locked to the other, as is the case between Pluto and Charon.
This effect is employed to stabilize some artificial satellites.

Fig. 1: Tidal locking results in the Moon rotating about its axis in about the same time it takes to orbit the Earth. (Source: Wikipedia)

Fig. 1, cont.: Except for libration effects, this results in the Moon keeping the same face turned towards the Earth, as seen in the figure
on the left. (The Moon is shown in polar view, and is not drawn to
scale.) If the Moon were not spinning at all, it would alternately
show its near and far sides to the Earth while moving around our
planet in orbit, as shown in the figure on the right.

Fig. 2: Lunar librations in latitude and longitude over a period of one month (Source: Wikipedia)

Libration is manifested as a slow rocking back and forth of the Moon
as viewed from Earth, permitting an observer to see slightly different
halves of the surface at different times.

There are three types of lunar libration:

Libration in longitude results from the eccentricity
of the Moon's orbit around Earth; the Moon's rotation sometimes leads
and sometimes lags its orbital position.

Libration in latitude results
from a slight inclination between the Moon's axis of rotation and the
normal to the plane of its orbit around Earth. Its origin is analogous
to how the seasons arise from Earth's revolution about the Sun.

Diurnal libration is a small daily oscillation due to the Earth's
rotation, which carries an observer first to one side and then to the
other side of the straight line joining Earth's and the Moon's
centers, allowing the observer to look first around one side of the
Moon and then around the other—because the observer is on the surface
of the Earth, not at its center.

Isn't saying "it is spinning at just the right rate" like saying "I don't fall to the centre of the Earth or fly off into space because Earth's surface pushes me up with just the right force to balance my weight"?
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David GarnerFeb 3 at 14:02

The period of rotation of the moon is ~27.322 days, and the period of revolution is also ~27.322. This means that for every degree it turns around the Earth, it turns a degree around itself, so the same side always faces us.

This is due to tidal forces coupling the various oscillators in the system (revolution of moon, orbit of moon, revolution of Earth). When oscillators are coupled, they have a tendency to settle to a state that is either in phase or 180 degrees out of phase*. Both cases give rise to tidal locking here.

This is an experiment you can try out suspend a pendulum from each end of a ruler, and give them a small phase difference. Over time, the phases will match.

As an complement to the other answers, let me address the question of why planets tend towards tidal locking. In short, the torque applied by the differential gravitational force between both sides of the surface of the planet induces friction, which in turn dissipates aways the excess spin of the (proto) moon when it is not tidally locked. When locking occurs the dissipation is minimized.
Another artifact is that the moon is also moving away from the earth (as it looses angular momentum). See e.g. http://curious.astro.cornell.edu/question.php?number=124

The other answers here are fantastic at explaining in a technical sense.

For an everyday example, imagine taking something asymmetrical--like a marble with some clay stuck to it--and spinning it. The asymmetry eventually brings the object to spin in a certain way. The moon is like that, except way more complicated, since its interaction with Earth is a part of the equation.

Could you explain the WHY of tidal locking in more detail ? Yours is the only answer that offers an explanation in this direction. I think that's the more important information here instead of just saying "Hey look, both the periods of its rotation and revolution are the same."
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Arjun J RaoAug 26 '14 at 10:08

RE: Always seeing the same side of the moon.
It may also be attributed to a difference in density of the moon. It was established recently that there are gravitational variations around the moon that would support the possibility that there is a higher density of material on one side of the moon. The higher density area of moon would then always face the earth. This would be similar to taping a heavy weight to one side of a ball on a hard surface. The heavier side of the ball will always face down towards the floor and towards the gravitational force. This is the same result of the heavier side of the moon facing towards the gravitational force emanating from the earth. It's too much of a coincidence that the moon rotates perfectly to always keep the same side facing towards the earth. Especially since the moon is moving away from the earth and is changing it's distance and time to revolve around the earth constantly. I came up with this theory about 10 years ago and I have not seen any other theories that would convince me that this is not a real possibility. This theory would also explain why Mercury always faces the same side towards the sun. There are no tides on Mercury to affect how it is orientated and it would appear that both bodies are affected by the same forces in keeping the same sides facing the same direction.

Can you give a credible reference for this "theory" to avoid a downvote? (including the idea that the mass of the moon has a significantly uneven distribution with respect to a plane at right angles to the Earth-Moon vector.) Also explain why there are no tides on Mercury (oh and Mercury does not keep the same face to the Sun).
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Rob JeffriesJan 16 at 15:56

What ever article that I read years ago about Mercury was incorrect. Apparently, at one time it was thought that Mercury might have one side always facing the Sun. The was proven to not be accurate. Though this does take away a secondary argument. That doesn't change my initial theory. The two gravitational sensing satellites that were sent around the moon did find irregularities in the gravitation from the moon. (See: en.wikipedia.org/wiki/Gravitation_of_the_Moon)
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Dan MitchellJan 18 at 4:03